REVIEW OF SCIENTIFIC INSTRUMENTS 85, 123114 (2014)

A force calibration standard for magnetic tweezers Zhongbo Yu, David Dulin,a) Jelmer Cnossen, Mariana Köber, Maarten M. van Oene, Orkide Ordu, Bojk A. Berghuis, Toivo Hensgens,b) Jan Lipfert,c) and Nynke H. Dekkerd) Department of Bionanoscience, Kavli Institute of Nanoscience, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

(Received 3 October 2014; accepted 27 November 2014; published online 23 December 2014) To study the behavior of biological macromolecules and enzymatic reactions under force, advances in single-molecule force spectroscopy have proven instrumental. Magnetic tweezers form one of the most powerful of these techniques, due to their overall simplicity, non-invasive character, potential for high throughput measurements, and large force range. Drawbacks of magnetic tweezers, however, are that accurate determination of the applied forces can be challenging for short biomolecules at high forces and very time-consuming for long tethers at low forces below ∼1 piconewton. Here, we address these drawbacks by presenting a calibration standard for magnetic tweezers consisting of measured forces for four magnet configurations. Each such configuration is calibrated for two commonly employed commercially available magnetic microspheres. We calculate forces in both time and spectral domains by analyzing bead fluctuations. The resulting calibration curves, validated through the use of different algorithms that yield close agreement in their determination of the applied forces, span a range from 100 piconewtons down to tens of femtonewtons. These generalized force calibrations will serve as a convenient resource for magnetic tweezers users and diminish variations between different experimental configurations or laboratories. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904148] INTRODUCTION

Advances in single-molecule instrumentation over the past two decades have resulted in high-resolution instruments capable of monitoring positions at the nanometer-scale with sub-second temporal resolution.1–3 These developments make it possible to examine the biophysical properties of in vitro enzymatic reactions4, 5 and to develop accompanying theoretical models.1 For example, it has become routine to monitor the progression of motor enzymes on a nucleic acid track at near-basepair resolution,6, 7 which makes it possible to unravel their underlying mechanochemistry. Magnetic tweezers are a versatile single-molecule technique1, 8, 9 that is capable of applying both forces and torques to tethered molecules. Magnetic tweezers can readily apply and measure forces in a wide range from >100 pN down to 1pN )

τmeasure ≈

8π 2 ηRξ l0 , kB T ε 2

(F < 1pN ) ,

or:

where τ measure is the minimum measurement time for a desired statistical accuracy ε (which we typically set to 0.05), η is the viscosity (0.001 Pa s for water), R is the bead radius, l0 is the DNA contour length, F is the force at a particular magnet position, and ξ is the DNA persistence length28 (equal to 47 nm under our experimental conditions of 10 mM Tris-HCl (pH = 7.4), 1 mM EDTA, and 100 mM NaCl).29 Magnet configurations and beads

Assembly of the surface-DNA-bead system

We employ a similar procedure to assemble a flow cell for magnetic tweezers as previously reported.26 First, we suspend 5 μl latex beads in ethanol (0.002% w/v, 3 μm diameter, Invitrogen, Life Technologies, Carlsbad) on a coverslip (MenzelGläser, Cat#: BB024060A1). Then, we heat the coverslip at 90 ◦ C for 5 min to melt the beads onto the surface. A volume of 5 μl nitrocellulose (0.1% w/v in ethanol) is added to the coverslip to improve the adsorption of anti-digoxigenin antibodies in a subsequent step. After drying this coated coverslip at 90 ◦ C for 5 min, we place a double-layer parafilm spacer containing a single channel on top. A second coverslip, pre-

F = kB T z/δx 2 ,

(1)

where kB is the Boltzmann constant, T is the absolute temperature, z is the DNA extension, and δx2  is the variance of the bead position. Repetition of such a measurement at various distances between the magnets and the magnetic beads then results in a complete calibration curve. In what follows, we make use of a ‘magnet distance’, which is defined such that the point where the bottom surface of the magnets would touch the lower inner surface of the flow cell is set to zero (Figure 1). To illustrate such a measurement, we display a number of traces of bead motion (Figure 2(a)). These traces were acquired for an M270 bead tethered to a 20.6 kbp DNA and pulled upon by a pair of vertically aligned permanent magnets (Materials and Methods section). The gap size between the

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FIG. 2. Force calibration in the time domain. (a) The x positions of an M270 bead recorded at acquisition frequencies of 0.06, 0.1, 0.12, 0.25, 0.5, 1, and 2 kHz (dark to light colors) for a magnet gap size of 0.5 mm and a magnet distance of 1.1 mm for a duration of 10.5 s. Traces are collected from the same DNA-tethered bead, and offset upwards for clarity. (b) Histograms of the traces in (a). A blurring effect is visible: lower acquisition frequencies and longer integration times result in narrower histograms. (c) Histograms of the x positions of an M270 bead after the blur-correction32 (details in main text). (d) Computed variances of the data in (b) before blur-correction (black) and of the corrected data in (c) (red). (e) Forces derived from the position variances as a function of magnet distance. Shaded area is inaccessible. (f) The errors in the forces in (e) relative to force values deduced directly from data obtained at 2 kHz. (g) Forces after the blur-correction. The shaded area is inaccessible due to the finite thickness of the flow cell. (h) The errors in the forces in (g) relative to that of data acquired at 2 kHz. The color code is the same in all panels apart from (d).

magnets equaled 0.5 mm, and the magnet distance was set to 1.1 mm. In principle, it would suffice to deduce the force from the variance of the transverse fluctuations (together with a measurement of the DNA extension z). However, finite acquisition frequencies (fs ) bias the measured variance of the fluctuation and, therefore, affect the force measurement of magnetic tweezers, due to the time-averaging of the fluctuations over the finite integration time (W ). The bias due to the finite acquisition time is particularly relevant when W is longer than the characteristic timescale (τ ) of the bead’s motion.32 We here examine the effect of finite acquisition frequencies by collecting traces at the acquisition frequencies of 0.06, 0.1, 0.12, 0.25, 0.5, 1, and 2 kHz, respectively, where corresponding camera W equals to W = 1/(2π fs ). In other

words, the camera shutter was continuously open (zero dead time) during the acquisition of an individual frame. The position histograms of traces recorded at different acquisition frequencies (Figure 2(b)) clearly demonstrate that longer shutter times result in reduced variances (quantified in Figure 2(d), black curve), and hence that simple computation of the variance does not provide a correct value for the applied force. These differences in measured variances are clearly undesirable, as they result in systematic errors in the measured forces. Fortunately, the bias due to finite acquisition times can be corrected, as illustrated by Wong and Halvorsen32 who introduced a motion blur correction function: 2 2 (2) S (α) = − 2 (1 − exp (−α)) , α α

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where α is the ratio of the camera integration time W to the characteristic timescale τ of a bead in a harmonic trap: α ≡ W/τ. Using the motion blur correction function, one can correct the measured variance var(Xm ) to obtain the true variance var(X): var(X) =

var(Xm ) . S (α)

(3)

This permits the correction of underestimates that result from the use of var(Xm ). Such underestimates are particularly significant at high integration times (e.g., 74% of var(X) for camera integration time W = τ ) and decrease for shorter integration times (e.g., 90% and 96.8% of var(X) for W = 1/3τ , 1/10τ , respectively). These computations indicate that, for the DNA tether length employed here, the use of the highest acquisition frequency (2 kHz) results in an accuracy of the force measurement by direct computation of the measured variance var(Xm ) that exceeds 90%. For reduced acquisition frequencies below 2 kHz, the use of Eq. (3) becomes imperative, and we can clearly observe the effect of the corrections on the histograms of bead positions (Figure 2(c)). The resulting values of var(X) plotted as a function of the camera acquisition frequency (Figure 2(d), red curve) are in good agreement with the direct computation of the variance from data acquired at 2 kHz. We plot the forces based on both the measured and corrected variances as a function of magnet distance (Figures 2(e)–2(g)). As expected, data acquired at low acquisition frequencies will result in overestimation of the true forces, with the relative errors being most pronounced in the high force regime where τ = γ /k is shortest (Figure 2(f)). For example, the relative error in the forces deduced from data acquired at 0.06 kHz (defined relative to forces deduced from data acquired at 2 kHz) increases from 0.2 to 2.5 as the magnets distance is decreased from 4.3 mm to 0.5 mm (Figure 2(f)). After correction, the relative errors are reduced to less than 0.1, provided that the acquisition frequencies exceed a lower bound (0.25 kHz) over the full range of magnet distances between 0.5 and 4.4 mm (Figure 2(h)). This lower bound can be decreased to 0.1 kHz if the magnet distances always exceed 2.4 mm. In other words, one can deduce accurate force values (to within 10%) from the analysis of bead positions in the time domain provided that the acquisition frequency exceeds certain limiting frequencies linked to the magnet distances employed. An alternative method to calibrate the forces in magnetic tweezers33 from bead positions in the time domain was proposed by Lansdorp et al. and relies on the computation of the Allan variance (Materials and Methods section). The Allan variance measures the signal stability over a given timescale to directly determine the uncorrelated noise and the magnitude of any drift.34 In the Allan variance algorithm, one first averages the bead position over a certain sampling time. The difference between two consecutive samples of bead position is used to derive the ensemble-averaged variance, which is twice the Allan variance. Using the same data sets as above, we have computed the forces versus magnet positions according to this method (Figure S2 in the supplemen-

tary material).41 The resulting forces display identical lower bounds on the acquisition frequencies as those deduced by employing the correction method introduced by Wong and Halvorsen.32 We note that the two preceding approaches to force calibration impose a lower bound on the acquisition frequency because we have fixed the camera exposure time at the inverse of the acquisition frequency. That is, the camera shutter is continuously open (zero dead time) and data are continuously averaged over this period. An alternative approach is to employ a lower camera speed with a reduced exposure time (non-zero dead time): for instance, one may acquire images at an acquisition frequency of 0.1 kHz and an exposure time of 1 ms (corresponding to 9 ms of dead time). Using such conditions, we have again measured forces, now on M270 beads tethered to the DNA pulled on by a pair of vertically aligned magnets separated by a gap size of 1 mm. Forces are computed from the variances of the bead positions in the time domain (Figure S3 in the supplementary material).41 Within experimental error, the resulting forces agree with those deduced from data acquired at 2 kHz under zero dead time conditions. In other words, provided that enough light remains for illumination, the blurring effect can be significantly suppressed through the imposition of a non-zero dead time on a low speed camera. However, a drawback of this approach is that it results, for the same number of frames acquired, in an increased acquisition time compared to the strategy of employing zero dead time. In addition to these approaches for correcting videoimage motion blur from data in the time domain, there are two approaches operating in the frequency domain that can be used to determine the applied forces in magnetic tweezers.21, 33 As before, one starts by recording a bead’s Brownian fluctuations, under conditions of zero camera dead time. One then makes use of the fact that bead motion in any dimension (the x dimension is selected here) in a medium with viscosity η can be described by a Langevin equation:33 kx + γ x˙ = FL ,

(4)

where k is the spring constant of the harmonic system, γ is the drag coefficient equal to 6π ηr, and FL is the Langevin force which obeys the fluctuation-dissipation relation: FL (t + t0 )FL (t) = 2γ kB Tδ(t0 ). We ignore the inertial force in the Eq. (4) because the friction occurs over an undetectable time interval, ∼10−6 s. Taking the magnitude of Fourier transform of this Langevin equation, one obtains the power spectral density (PSD) of bead motion as a function of frequency:33 P (f ) =

2π 2 γ

k T , B 2 fc + f 2

(5)

where fc is a cut-off frequency equal to k/2π γ . We use twosided power spectra throughout, so that integrating P(f) over the range (−∞, +∞) yields x2  = kB T/k. PSDs for the same dataset as in Figure 2(a) are plotted in Figure 3(a). As expected, the PSDs for datasets acquired at higher acquisition frequencies extend out to higher frequencies (compare the dataset acquired at 2 kHz to the datasets acquired at 1, 0.5, 0.25, 0.12, 0.1, and 0.06 kHz, Figure 3(a)), as do the corresponding fits to Lorentzian functions (shown in red). The data comprising such a PSD are, as before, subject to the

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Rev. Sci. Instrum. 85, 123114 (2014)

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FIG. 3. Force calibration in the spectral domain. (a) Power spectra converted from the x positions in Figure 2(a) (identical color code) over the time interval of 10.5 s. The range of the spectra in the high frequency domain is limited by the acquisition frequencies employed. Fits of the spectra to Lorentzian functions are shown in red. Traces are offset upwards for clarity. (b) Trap stiffness versus acquisition frequency as deduced from the fits in (a). (c) Corner frequencies deduced from data acquired at different acquisition frequencies plotted as a function of magnet distance. Horizontal lines represent the Nyquist frequencies of 30, 50, and 60 Hz. (d) Forces derived from the PSD method described by te Velthuis et al.21 as a function of magnet distance. The forces deduced at the acquisition frequency of 0.06 kHz show large fluctuations at low magnet distances. (e) The errors in the forces in (d) with respect to data obtained at 2 kHz. The results from data acquired at 0.06 kHz are not shown for clarity. The color code is the same in all panels apart from (b). The results above are deduced from the same DNA-tethered bead.

distorting effects of low-pass filtering and aliasing,21, 33 resulting in a biased measurement, P˜ (f ). Previously, te Velthuis et al. have discussed21 how to recover the underlying true PSD by fitting the integral of the power spectrum with an iterative and approximate correction. Using such an approach (denoted “te Velthuis PSD” in what follows) on data acquired for an M270 bead tethered to the DNA and pulled on by magnets at a distance of 1.1 mm, the computed trap stiffness (kx ) remains relatively constant for camera acquisition frequencies fs exceeding 0.25 kHz (Figure 3(b)). Alternatively, for a camera acquisition frequency fixed at 0.1 kHz, the cut-off frequency that can be extracted from the datasets increases exponentially with magnet distance until fc > 40 Hz (Figure S4 in the supplementary material).41 When the magnet distance 16 pN), the best approach is to acquire data at high acquisition frequencies (e.g., 2 kHz), since no further data correction is required. If such a camera is not available, we recommend the use of Wong’s motion blur correction function on data in the temporal domain. For magnet distances larger than 2.4 mm (corresponding to forces < 16 pN), more choices are available. Acquiring data at high acquisition frequencies (e.g., 2 kHz) remains a good option, although the collection of extensive datasets in this regime may challenge either the control software or the computer hardware. In this distance limit, our results illustrate that the errors in camera acquisition at low frequencies coupled with long integration time can be reliably corrected, irrespective of the algorithm employed, provided that the camera acquisition frequency exceeds 0.1 kHz. Thus, given their simplicity, we recommend the PSD-based methods for force calibration at magnet distances exceeding 2.4 mm. Matlab-based implementations of all these force calibration codes are available.41 To illustrate the use of these distinct approaches (analysis in either the time domain or the spectral domain), we calibrate forces on M270 beads over the full range of magnet distances between 0.5 and 10.4 mm. We choose to assemble separately acquired data at high forces (corresponding to magnet distances between 0.5 and 4.4 mm; acquired at a camera acquisition frequency of 2 kHz and analyzed in the time domain) and low forces (corresponding to magnet distances between 2.4 and 10.4 mm; acquired at a camera

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FIG. 5. Force calibration curves for M270 beads under four magnet gap sizes. (a) Force calibration curve assembled from two sections: the low force region in black squares (magnet distance from 2.4 to 10.4 mm, Nbeads > 10) recorded at 0.10 kHz and the high force region in red circles (magnet distance from 0.5 to 4.4 mm, Nbeads > 5) recorded at 2 kHz. The magnet gap size equals 0.5 mm. The green solid line represents the fit to a double exponential function. (b) Log-lin plot of the data in (a). (c) Force calibration curves of M270 beads (Nbeads > 5) for four magnet gap sizes: 0.3 mm (blue diamonds), 0.5 mm (green squares), 1 mm (red circles), and 2 mm (black triangles). Solid lines represent the fit to a double exponential function. The inset zooms into the high force region. (d) Log-lin plot of the data in (c). Symbols represent the average of forces, and error bars indicate the standard deviations. The shaded area is inaccessible due to the finite thickness of the flow cell.

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TABLE I. Double exponential fitting results from force-magnet distance curves. All forces were determined using the PSD method with correction of blurring and aliasing effects.21 The units of force are in piconewtons. The units of the magnet distances and the gap sizes are in millimeters. Index 1 2 3 4 5 6 7 8

Fit to a double exponential function

Magnet distance (mm)

Bead

Gap (mm)

F(Zmag ) = −0.0078 + 43.5exp( − (Zmag − 0.4)/1.58) − 30.4exp( − (Zmag − 0.4)/0.319) F(Zmag ) = −2.74 + 71.8exp( − (Zmag − 0.4)/1.33) + 4exp( − (Zmag − 0.4)/36.9) F(Zmag ) = −0.0029 + 42.5exp( − (Zmag − 0.4)/0.49) + 62.8exp( − (Zmag − 0.4)/1.44) F(Zmag ) = −0.019 + 60.4exp( − (Zmag − 0.4)/0.563) + 56.9exp( − (Zmag − 0.4)/1.46) F(Zmag ) = −0.0028 + 5.69exp( − (Zmag − 0.4)/1.53) − 3.07exp( − (Zmag − 0.4)/0.351) F(Zmag ) = −0.117 + 8.68exp( − (Zmag − 0.4)/1.25) + 0.2exp( − (Zmag − 0.4)/14.5) F(Zmag ) = −0.0009 + 5.29exp( − (Zmag − 0.4)/0.52) + 7.16exp( − (Zmag − 0.4)/1.39) F(Zmag ) = −0.0009 + 6.21exp( − (Zmag − 0.4)/0.427) + 7.27exp( − (Zmag − 0.4)/1.35)

0.7–10.4 0.5–10.4 0.5–10.4 0.5–10.4 0.5–10.4 0.5–10.4 0.5–10.4 0.5–10.4

M270 M270 M270 M270 MyOne MyOne MyOne MyOne

2 1 0.5 0.3 2 1 0.5 0.3

acquisition frequency of 0.1 kHz and analyzed in the spectral domain). The resulting force calibration curve for a magnet configuration with a gap size of 0.5 mm (Figure 5(a)) spans three orders of magnitude in force and shows excellent agreement in the overlapping region, i.e., magnet distances between 2.4 and 4.4 mm (log-lin plot of the same data shown in Figure 5(b)). Note that in this plot, the points correspond to the averages over several beads, with the error bars reflecting the corresponding standard deviations. For M270 beads, the variation between beads contributes to an uncertainty of 7%, which is within the range of error that is commonly accepted in force measurements.17, 19, 35 This approach can be expanded to include magnet configurations with gap sizes of 2.0, 1.0, and 0.3 mm, which allows us to access an even larger range of forces (Figure 5(c); log-lin plots of the same data shown in Figure 5(d)). One can clearly note that the maximum force increases as the gap size between the magnets is reduced from 2 to 0.3 mm (Figure 5(c), inset). The maximum force measured equals ∼117 pN for magnets separated by a gap size of 0.3 mm and M270 beads. We additionally validate our force measurements using biological markers, e.g., the characteristic worm-like chain behavior of dsDNA36 (Figure S6 in the supplementary material41 ). Additionally, B-form dsDNA undergoes a characteristic phase transition at an applied force of ∼65 pN36–39 in which its extension increases by ∼70%. Using our force calibration curves, we find that this over-stretching transition occurs at 65 ± 5 pN in Tris-EDTA buffer (pH = 7.4) supplemented with 100 mM NaCl. In all cases, we fit the resulting force calibration curves as a function of magnet distance to a double exponential function:40 F (z) = δ + α0 exp(−z/ζ0 ) + α1 exp(−z/ζ1 ), where z is the magnet position, F is the force, and δ, α 0 , ζ 0 , α 1 , ζ 1 are fitting parameters. The resulting fits are summarized in Table I. To enhance the generality of our calibration, we perform force measurements on MyOne beads using magnets separated by the same four gap sizes (0.3, 0.5, 1.0, and 2.0 mm). All other parameters are identical. Since MyOne beads have a lower magnetic moment than M270 beads, the characteristic frequency of their tethers (fc = κ/2π γ ) is reduced compared to that of M270-based tethers at the same magnet distance. Hence, an acquisition frequency of 0.1 kHz suffices to collect the full dataset. The resulting forces as a function of magnet distance display similar trends to those observed for

the forces on M270 beads, with comparable uncertainty (8%) arising from bead-to-bead variations, but with much lower maximum applied forces, ∼14 pN (Figure S7 in the supplementary material).41 The final fitting results of the double exponential functions are summarized in Table I. Our laboratory has recently developed a novel bead tracking software package23 (freely available at nynkedekkerlab.tudelft.nl) that employs a combination of Labview, C++ and CUDA to enable the parallel tracking of multiple beads (e.g., tracking 1000 beads at an acquisition frequency of 20 Hz) or high speed tracking of beads (e.g., tracking two beads at an acquisition frequency of 10 kHz) in magnetic tweezers. In this work, we have employed this software package to collect all datasets. Under the conditions of the DNA tethered M270 beads, a pair of vertically aligned magnets with 1 mm gap size, and an acquisition frequency of 2 kHz at high forces and 0.1 kHz at low forces, the resulting calibration curve agrees well with that obtained by the predecessor Labview package (Figure S8 in the supplementary material),41 which has been used in a series of published works.19, 21, 26 The present force calibration thus validates the newly developed package. In addition, we have cross-validated these force calibrations on four different magnetic tweezers instruments, which reveals excellent agreement (Figure S9 in the supplementary material).41

CONCLUSIONS

Magnetic tweezers have become a popular and robust technique to measure the forces applied to or generated by biological molecules. To provide detailed insight into the force measurement and facilitate standardization of the conventional permanent magnet-based magnetic tweezers, we have presented a complete set of calibrated look-up tables of the achievable forces on two different types of beads for four alternative magnet configurations. The achievable forces range from more than 110 pN down to 8 fN, while the force calibrations show excellent consistency on four independent magnetic tweezers instruments. We anticipate that the generalized force calibrations demonstrated here will not only serve as convenient look-up tables for any user but also help to limit experimental variations from instrument to instrument.

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ACKNOWLEDGMENTS

We thank Richard Janissen and Jacob Kerssemakers for valuable discussions. We acknowledge funding to NHD from the Netherlands Organisation for Scientific Research (NWO) via a VICI grant as well as from the European Research Council via a Starting Grant. 1 I.

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A force calibration standard for magnetic tweezers.

To study the behavior of biological macromolecules and enzymatic reactions under force, advances in single-molecule force spectroscopy have proven ins...
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